A-Math 2023 Worked Solutions

Line-and-curve tangency

3 marks · algebra (tangent condition)

Partial fractions with a repeated linear factor

5 marks · algebra (partial fractions, repeated linear factor)

Pythagorean identities

5 marks · trigonometry (prove an identity)

Product rule with e^-2x

7 marks: 3 + 4 · calculus (product rule, then integrate by reversing)

Solve a trig equation

6 marks · trigonometry (solve over a range)

Quotient rule

6 marks: 2 + 4 · calculus (quotient rule; decreasing interval)

Always-negative condition

6 marks · algebra (quadratic always negative)

Simultaneous equations (one linear

5 marks · algebra (line meets curve, simultaneous)

Stationary points

8 marks: 4 + 4 · calculus (stationary points; maximum gradient)

Equation of a circle (complete the square)

10 marks: 4 + 2 + 4 · coordinate geometry (circle; dividing a line)

Integrate k/2t+1 to a logarithm

9 marks: 1 + 4 + 4 · calculus (rates of change)

Completing the square (vertex)

10 marks: 1 + 3 + 2 + 1 + 3 · algebra (projectile model, completing the square)

Perpendicular bisector

10 marks: 3 + 7 · coordinate geometry (perpendicular bisector)

Binomial general term

5 marks · algebra (binomial: coefficient of x^3)

Tangent-chord (alternate segment)

9 marks: 4 + 5 · geometry (similar triangles; tangent and semicircle)

Exponential model

9 marks: 1 + 3 + (2 + 3) · algebra (exponential-decay models)

Double-angle identity

10 marks: 5 + 5 · trigonometry (exact values; no calculator)

Differentiate 3x

10 marks: 7 + 3 · calculus (second derivatives, match coefficients)

Factor theorem

12 marks: 5 + 5 + 2 · algebra (factor theorem; exponential equation)

Amplitude and period

11 marks: 2 + 3 + 6 · trigonometry & kinematics

Area of isosceles triangles

12 marks: 3 + 2 + 2 + 5 · trigonometry (area, r-formula)

Model time = distance/speed

12 marks: 3 + 6 + 3 · calculus (minimisation)